The Role of Point-Fibredness in Iterated Function Systems: Exposition, Exploration, and Extensions
摘要
The classical Banach–Hutchinson–Barnsley framework for Iterated Function Systems (IFSs), based on the Banach fixed-point theorem applied to a system of strict contractions, establishes a foundational correspondence between fixed-point theory and fractal geometry. Generalizations of this framework often proceed via the application of advanced fixed-point theorems to the associated Hutchinson–Barnsley operator on hyperspaces and the corresponding Markov operator on measure spaces. An alternative approach, however, is based on the concept of a code map, which constructs a semiconjugacy between the IFS and a symbolic dynamical system defined over a shift space. This method facilitates to transfer ergodic and invariant measure-theoretic properties from the symbolic space to the attractor of the IFS. Although the coding-map approach appears sporadically in the literature, its potential as a unifying tool for the study of non-contractive IFSs has not been fully explored. In this note, we revisit and expand the coding-map framework, emphasizing its structural and analytical utility in the study of IFSs beyond the scope of classical contractions. We propose a unified viewpoint that accommodates a broader class of systems—possibly non-contractive—while retaining tractable dynamical and ergodic properties. Our focus is particularly on the existence and uniqueness of invariant measures, ergodicity, and the structural interplay between symbolic dynamics and geometric realizations. In doing so, we position the coding-map approach as a complementary and potentially unifying tool within the broader Hutchinson–Barnsley fractal framework.